Washing and wringing machines as they were widely called back in the early 17th century, probably appeared before the 1600’s in some form, but the first patent issues for such a device was awarded in England in 1691. Although I haven’t been able to find the name on the patent, and thus give you the inventor’s identity, there are some other milestones in the development of the modern washing machine that could also contend for the title since it depends on what you consider a true washing machine.
A sketch of a hand powered device appeared in the 1752 of “The Gentleman’s Magazine” a British publication, again with no known designer. In Germany, Jacob Schaffer published a design in 1767, which was the earliest name I could track down connected to the device.
If you consider a washing machine to be a device with a rotating drum, like a modern top-loader, than the first recorded device was invented in England in 1782 by Henry Sidgier. This seems to be the first mass-marketed machine, as it was sold around England as early as the 1790’s.
A “dryer” or ringer which squeezed water out like a primitive spin cycle was devised in 1862 by Richard Lansdale. If you consider a washing machine to be a device that cleans clothes and then spins the excess water out of them, then this would be the earliest known machine.